Entire functions sharing one small function

The uniqueness problem of entire functions sharing one small function was studied. By Picard’s Theorem, we proved that for two transcendental entire functions f (z) and g(z), a positive integer n≥9, and a(z) (not identically eaqual to zero) being a common small function related to f (z) and g(z), if f n(z)(f(z)-1)f′(z) and gn(z)(g(z)-1)g′(z) share a(z) CM, where CM is counting multiplicity, then g(z) ≡ f (z). This is an extended version of Fang and Hong’s theorem [ Fang ML, Hong W, A unicity theorem for entire functions concerning differential polynomials, Journal of Indian Pure Applied Mathematics, 2001, 32 (9): 1343-1348].     

亚纯函数与其微分多项式的值分布

通过研究亚纯函数的Nevanlinna值分布理论问题,并结合亚纯函数的小函数,及其微分单项式和微分多项式,得到一比较有趣的关于亚纯函数的计数函数密指量和微分多项式的不等式,此不等式改进了Fang,Yang及I Lahiri和S.Dewan等学者的结果。     

具有一个非零公共值的亚纯函数的微分多项式唯一性

证明了非常数的亚纯函数的一类非线性微分多项式具有一个非零公共值的亚纯函数的唯一性,其文结果改进了杨重骏和华歆厚的结果,扩充了方明亮的结果。     

亚纯函数与其微分多项式具1IM公共值

得到亚纯函数与其微分多项式具１ＩＭ 公共值的唯一性定理,推广了某些已知结果．     

Entire functions sharing a value with its weight

Introduction Let f and g be two meromorphic functions defined in the open complex plane C, and k be a nonnegative integer or infinity. For {}aC违U, denote by (;)kEaf the set of all a-points of f where an a-point of multiplicity m is counted m times if mk and otherwise k+1 times. If (;)(;)kkEafEag=, then, f and g are considered to share the value a with weight k, which is expressed by f and g share (a ,k). Clearly if f and g share (a ,k), then they share (a ,p) for all integer p in 0pk#. It…     

Normal families of meromorphic functions

A normal theorem concerning meromorphic functions sharing values was proved with the method of Zalcman- Pang.The theorem is as follows. If for each f in F, all zeros of f-a have multiplicity at least k (k≥2), f and its k-th derivative function share a, and if f=b whenever its k-th derivative equal b, then F is normal in D. This theorem improved the result of Chen and Fang [Chen HH, Fang ML, Shared values and normal families of meromorphic functions, Journal of Mathematical Analysis and Applications, 2001, 260: 124-132].     

Uniqueness of entire functions concerning weighted sharing

The uniqueness problem of entire functions concerning weighted sharing was discussed, and the following theorem was proved. Let f and 8 be two non-constant entire functions, m, n and k three positive integers, and n〉2k＋4. If Em（1,（f^n）^（k））= Em（1,（g^n）^（k））, then either f（z）=c1c^cz and 8（z）= c2c^cz or f=ts, where c, c1 and c2 are three constants satisfying （-1）^k（c1c2）^n（nc）^2k=], and t is a constant satisfying t^n=1. The theorem generalizes the result of Fang [Fang ML, Uniqueness and value sharing of entire functions, Computer ＆ Mathematics with Applications, 2002, 44： 823-831].     

复系数多项式凸组合的稳定性检验

两个给定的同次复系数多项式是Hurwitz稳定的,它们的凸组合是否也是Hurwitz稳定的?本文就此给出了一个检验方法.该方法以结式理论为依据,并应用了Sturm定理,不需求解多项式的根,仅经过有限步的运算就可完成对凸组合的稳定性判别,应用方便.     

Helmhoz方程的Legendre展开方法

A spectral method based on the Legendre polynomials for solving Helmholz equations was proposed. With an explicit formula for the Legendre polynomials in terms of arbitrary order of their derivatives, the successive integration of the Legendre polynomials was represented by the Legendre polynomials. Then the method was formulized for secondorder differential equations in one dimension and two dimensions. Numerical results indicate that the suggested method is significantly accurate and in satisfactory agreement with the exact solution.     

Hermite多项式的两种微分表示之恒等性的证明

用两种方法证明了Hermite多项式的两种微分表示是恒等的。     

结式的若干应用

结式是多项式理论中的一个重要概念.本文从7个方面阐述了结式的应用,包括判断2个多项式的公共零点,判定多项式是否有重根,计算多项式的判别式,寻找代数数满足的方程,平面有理曲线的隐式化,非线性代数方程组求解和不等式机器证明中半代数系统边界曲面的获得等.     

达布定理与罗尔定理的等价性

给出了微分学中达布定理与罗尔定理等价性的证明,并且获得了不用费马定理而用实数的连续性定理和导数定义证明这两个定理的一个方法。     

L~ψ空间中的Jackson定理

研究了Lψ空间中的函数用多项式逼近的问题 ,证明了Jackson定理 .     

不等式研究的几个新结论和新问题

讨论了带条件约束的差分代换证题方法和三角形几何不等式成立的2个重要条件;验证了轮换对称多项式基本定理猜想;得到了若干不等式新结论;提出了5个不等式新问题。     

Meromorphic functions sharing one value by weight

The uniqueness of meromorphic fuctions sharing one value was studied. Using the concept of weighted sharing, we proved the following theorem. For two meromorphic functions f and g which are not polynominals of degree less than a positive integer k, if f nf (k) and g ng (k) share (1,2), where n is another positive integer not less than k 10, then f nf (k) identically equals g ng (k) or f nf (k)g ng (k) identically equals 1. Particularly for k =1, we improved the results of Yang [Yang CC, Hua XH, Uniqueness and value-sharing of meromorphic functions, Annales Academi? Scientiarum Fennic? Mathematica, 1997, 22: 395-406], and Fang [Fang ML, Hua XH, Entire function that share one value, Journal of Nanjing University, 1996, 13(1): 44-48. (In Chinese)].     

二阶微分方程解的导函数比较定理

利用文中建立的两个微分恒等式研究了两类非线性微分方程解的导函数的零点比较定理及振动比较定理，推广了一些已知结论。     

Roger公式蕴含_6φ_5和式定理

本文证明关于q-超球多项式的Rogers公式蕴含着基本超几何函数论中的6φ5求和公式     

正十二面体的旋转群诱导出的置换群的轮换指标

目的：给出正十二面体的旋转群所诱导出的置换群的轮换指标.方法：通过分析所诱导出的置换群中元素的置换类型,利用Burnside定理和Polya定理给出相应的轮换指标多项式.结果：给出了所诱导出的6种置换群的循环指标多项式.结论：所给出的循环指标多项式可以为正十二面体的着色计数问题提供计算依据.     

多项式的区间稳定性

本文证明了多项式区间稳定性定理 ,并给出稳定区间的计算机程序及应用     

微分中值定理和定积分中值定理的相关性

探究了微分中值定理和定积分中值定理的关系,发现二者具有密切的联系,并给出了该相关性产生的原因.